Quantum mechanics

Theoretical physics

Phase (matter)

Pure mathematics

Mathematical physics

Boundary (topology)

Bloch wave

Boundary value problem

Eigenvalues and eigenvectors

Mathematical optimization

Acoustics

Hamiltonian (control theory)

Lattice (music)

Physics

Periodic boundary conditions

Mathematics

Chern class

Condensed matter physics

Phase diagram

Mathematical analysis

Quantum Hall effect

Electron

Hermitian matrix

Physical Review Letters

The relation between chiral edge modes and bulk Chern numbers of quantum Hall insulators is a paradigmatic example of bulk-boundary correspondence. We show that the chiral edge modes are not strictly tied to the Chern numbers defined by a non-Hermitian Bloch Hamiltonian. This breakdown of conventional bulk-boundary correspondence stems from the non-Bloch-wave behavior of eigenstates (non-Hermitian skin effect), which generates pronounced deviations of phase diagrams from the Bloch theory. We introduce non-Bloch Chern numbers that faithfully predict the numbers of chiral edge modes. The theory is backed up by the open-boundary energy spectra, dynamics, and phase diagram of representative lattice models. Our results highlight a unique feature of non-Hermitian bands and suggest a non-Bloch framework to characterize their topology.

Non-Hermitian Chern Bands